Finite Group Databasebeta

← #30640904

Subgroups — #30640904

Order 16256.

120 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550276C211yesmaximal, characteristic, complemented
2#22549993C211yesmaximal, characteristic
3#22549699C211yesmaximal, characteristic, complemented
4#22549698C211yesmaximal, characteristic, complemented
5#22549696C211yesmaximal, characteristic, complemented
6#22549686C211yesmaximal, characteristic
7#360×C127C211yesmaximal, characteristic
8#16861216—21no—
9#16861200—21no—
10#16861149C2211yescharacteristic
11#16861145C2211yescharacteristic
12#16861142—21no—
13#16861141C2211yescharacteristic
14#16861136—21no—
15C8×Dic127C2211yescharacteristic
16#135×C127C2211yescharacteristic
17M32×C127C2211yescharacteristic
18#106×C127C2211yescharacteristic
19#14523232—21no—
20#14523232—41no—
21C2×C127⋊C8—41no—
22C2×C127⋊C8D411yescharacteristic
23C4⋊Dic127—21no—
24C4⋊Dic127—41no—
25Dic127⋊C4—41no—
26C4×Dic127D411yescharacteristic
27C127⋊C16—21no—
28#41×C127—21no—
29Q16×C127Q811yescharacteristic
30SD16×C127—21no—
31D8×C127Q811yescharacteristic
32C2×C8×C127C2311yescharacteristic
33C4⋊C4×C127—21no—
34C16×C127—21no—
35C2×Dic127—21no—
36C2×Dic127—41no—
37C127⋊C8—41no—
38C127⋊C8SD1611yescharacteristic
39C127⋊C8SD1611yescharacteristic
40Q8×C127—21no—
41D4×C127—21no—
42D4×C127—41no—
43C2×C4×C127—41no—
44C2×C4×C127—41no—
45C2×C4×C127C2×D411yescharacteristic
46C8×C127C2×Q811yescharacteristic
47C8×C127#4111yescharacteristic
48C22×C127—41no—
49C22×C127AGL(1,ℤ/8)11yescharacteristic
50C4×C127—41no—
51C4×C127—81no—
52C4×C127C22⋊Q811yescharacteristic
53C4×C127C2×SD1611yescharacteristic
54Dic127—41no—
55Dic127—81no—
56C2×C127—21no—
57C2×C127—81no—
58C2×C127#47611yescharacteristic
59#2231—1271nomaximal, Sylow
60C127#223111yescharacteristic, complemented, Sylow
61#499—1271no—
62#443—1271no—
63#365—1271no—
64#365—1271no—
65#364—1271no—
66#360—1271no—
67#360D12711yescharacteristic
68#135C2×D12711yescharacteristic
69#131—2541no—
70#125—2541no—
71M32—1271no—
72M32C2×D12711yescharacteristic
73#108—1271no—
74#107—2541no—
75#106—1271no—
76#106C2×D12711yescharacteristic
77#104—2541no—
78C4×C8—1271no—
79#41—21no—
80Q16Dic25411yescharacteristic
81SD16—21no—
82D8Dic25411yescharacteristic
83M16—5081no—
84M16—2541no—
85C2×C8—1271no—
86C2×C8—5081no—
87C2×C8C22×D12711yescharacteristic
88C4⋊C4—21no—
89C4⋊C4—5081no—
90C4⋊C4—5081no—
91C4⋊C4—2541no—
92C42—1271no—
93C16—21no—
94C16—2541no—
95Q8—21no—
96D4—21no—
97D4—41no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—5081no—
101C2×C4—2541no—
102C2×C4D4×D12711yescharacteristic
103C8—1271no—
104C8—1271no—
105C8—5081no—
106C8C2×Dic25411yescharacteristic
107C8#1452324711yescharacteristic
108C22—41no—
109C22#1686123311yescharacteristic
110C4—41no—
111C4—81no—
112C4—10161no—
113C4—5081no—
114C4#1686118611yescharacteristic
115C4SD16×D12711yescharacteristic
116C2—21no—
117C2—81no—
118C2#2255016411yescharacteristic
subgroupG/H
centerC2#22550164
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#360×C127C2
socleC2×C127#476
prime radical#135×C127C22
hypercenter#360D127
hypocenterC127#2231
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22550276C2semidirect
3. #22549699C2semidirect
4. #22549698C2semidirect
5. #22549696C2semidirect
60. C127#2231semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225502762
#225499932
#225496992
#225496982
#225496962
#225496862
#360×C1272
#2231127